Digital Signal Processing Reference
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A simple and elegant algorithm has been proposed in [84] to find an approximate solu-
tion to (8.103) using a properly rescaled weight vector of the original nonrobust problem
(8.100) or its convex relaxed reformulation presented in [83]. Another robust technique
that solves (8.103) has been developed in [85] using the convex SDP approach.
8.4.3 Multicast Transmit Beamforming
The multicast transmit beamforming problem is a natural, yet nontrivial, extension of
the broadcast problem to the case when different data streams have to be transmitted to
different groups of users.
Let us modify the broadcast scenario by assuming a total of
I
(1 ≤
I
≤
L
) multicast
groups {D
1
, …, D
I
}, where
D
i
is the index set of receivers participating in the
i
th
group
[86, 87]. It will be assumed that no user can be shared by more than one group, that is,
D
i
∩ D
k
=
∅
for any
i
≠
k
and D
1
∪ D
2
∪
…
∪ D
I
= {1, 2, …,
L
}, which also implies that
I
∑
=
1
D
i
L
.
i
=
Using these conventions, the joint design of the transmit beamformers for multiple
user groups amounts to minimizing the total transmit power subject to the QoS con-
straints [86]:
2
H
wh
k
m
H
min
{}
ww
subjectto
≥
γ forall
m
∈ , , =, ,
D
k l
1
…I
, (8.104)
i
i
∑
m
k
2
I
2
H
w
σ
+
wh
i i
=
1
k
l
m
l
≠
k
where
w
i
is the weight vector used for the
i
th
group and, as before, γ
k
and
h
k
are the mini-
mal acceptable QoS and the downlink channel vector of the
k
th
user.
Clearly, as (8.104) represents a generalization of (8.100), it it also NP-hard. Moreover,
in contrast to (8.100), problem (8.104) can be infeasible due to crosstalk-type interfer-
ence between the user groups. Using an approach similar to that used in [83] for relaxing
broadcast problem (8.100) to a convex form, the authors of [86] have derived a relaxation
of (8.104) to the following convex SDP problem:
I
∑
∑
+
γσ
2
min
tr
{
Z
}
subjecttotr
{
Q
Z
}
≥
γ
tr
{
QZ
}
i
m
k
m
ml
mm
i
I
{
Z
=
1
i
=
1
l
≠
k
forall
m
∈, ,=, , ,
D
k l
1
…I
(8.105)
k
H
Z
=,
ZZ
0
forall
i
=, ,.
1
…I
i
i
i
Similar to (8.101), problem (8.105) can be directly solved using available convex optimi-
zation tools [50].
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